While messing around with the idea of ordinal collapsing functions, I stumbled upon an interesting simple function:
$$C(0)=\{0,1\}\\C(n+1)=C(n)\cup\{\gamma+\delta:\gamma,\delta\in C(n)\}\\\psi(n)=\min\{k\notin C(n),k>0\}$$
The explanation is simple. We start with $\{0,1\}$ and repeatedly add it...